Prove QQ16 final-only loss despite 64 initial query coordinates

research/qq16-final-record-obstruction/PROOF.md

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+# The final records do not recover the recorded first-pair cut
+
+This is a scoped theorem for the actual routed QQ16 maps, on the retained
+ordinary-bank schedule with no added selected intervention before the
+completion of F2. It upgrades Philippe Van Velzen's finite-model
+final-only failure to a regular-cover argument. It does not contradict
+PR #2174: that recovery theorem explicitly grants the selected state at
+the cut after B1, whereas the observation contract here grants only the
+final selected state.
+
+The result is not a claim about every selected controller. In particular,
+this package does **not** identify an unprovided historical first-six-swap
+selected program with the ordinary-only bridges used below. A swap or
+other intervention before F2 changes this theorem's hypotheses.
+
+## 1. Exact contract and conclusion
+
+Let G be the finite actual owner cover group, N=|G|. Work with
+rational-valued logical fields on G, constant in the payload coordinates.
+The actual cover maps preserve this subspace: their payload translations
+act trivially, while their right owner actions remain genuine regular
+actions. One may use the subgroup generated by the actual route matrices
+or a full owner group containing it with its regular action; the latter
+decomposes into free orbits. This proof does not apply to an arbitrary
+nonregular quotient action. It applies at every positive tensor payload
+size. Invertible native input/output frames transport the logical kernel
+family to the corresponding native interface; those frames need not
+preserve the payload-constant subspace themselves.
+
+Use the retained local forward and conserved backward maps from
+`research/joint-query/evidence/CONSERVED-QQ16.md`, equations (1)--(3).
+Write
+
+    B_s = R_s iota_U,
+    P_(s,A) = D_s^-1 iota_A R_A^-1,
+    ell_(s,A) = R_A pi_A D_s,
+
+and similarly for T. Selected sections and their selected readers are
+mutual inverses on their eight selected fields. U has 96 ports; A and T
+are the eight-port sets in the source packet.
+
+The exact 19-event recipe is read from the source-bound selected archive.
+The local calls at stages 0,...,8 use pairs
+
+    0, 0, 0, 1, 1, 2, 2, 3, 3.
+
+Even stages are forward F_s; odd stages are backward B_(s,s+1).
+An ordinary bridge `src=i,dst=j,coefficient=c` acts only on U:
+
+    X_j,U += c X_i,U;
+    Y_i,U -= c Y_j,U.
+
+Between local calls there are no additional selected operations in the
+base contract. The final ordinary terminal is `(X_U,Y_U) -> (Y_U,-X_U)`.
+Any homogeneous selected terminal, including the same signed exchange,
+has the same zero selected output on the family below.
+The named final Y fields are the completed-X descendants; named final X
+fields have the former-Y type.
+
+The observation O consists of:
+
+1. Every final selected X and Y field of all four pairs.
+2. Every final completed-X ordinary field, namely final Y_(k,U).
+3. All five forward query histories f0,f2,f4,f6,f8.
+4. The following eight initial ordinary-query records, each of dimension
+   eight per owner (64 coordinates per owner in total):
+
+       pair 0: B0 X_(0,U), B2 X_(0,U);
+       pair 1: B3 X_(1,U), B4 X_(1,U);
+       pair 2: B5 X_(2,U), B6 X_(2,U);
+       pair 3: B7 X_(3,U), B8 X_(3,U).
+
+These are exactly the bank-local initial records used in the credited
+four-bank final-only experiment. Their storage and reads are granted
+without cost for this impossibility statement. No post-B1 selected cut,
+former-Y ordinary endpoint, or additional ordinary read is observed.
+
+**Theorem.** If J selects the original A/T inputs of pair 0, then
+
+    dim J(ker O) >= ceil(8N/7) > 0.
+
+Consequently no decoder, linear or otherwise, can recover those original
+selected inputs from this observation contract. In particular, these
+final records do not imply the post-B1-cut premise of PR #2174. This is
+an information obstruction for this fixed contract, not a native cost
+bound, a failure of the selected supplier, or a new kappa claim.
+
+## 2. Actual source-bound trace identities
+
+The checker reconstructs the centre matrix R from the original scalar
+program before its centre cut, and compares it with the retained QQ16
+packet. It checks
+
+    B_s B_s* = 48 I_(8N),
+    ||r_p||^2 = 4  for every ordinary port p,
+
+where * is the transpose and inverse regular action. The first identity
+also follows directly from `R_U R_U^T=48 I_8`: in each same-port term the
+route cancels with its inverse.
+
+For each of the six distinct stage pairs
+
+    (0,1), (0,2), (1,2), (2,3), (2,4), (3,4),
+
+the checker reconstructs all 96 actual relative owner-route matrices
+and verifies that they are pairwise distinct. Routes come from the
+73-dimensional DATA-PHASES source, and all 1008 reconstructed stage/port
+routes are matched to owner matrices at their exact stage/port positions in the
+native forward programs (stage zero has its separately checked identity
+route base case). They are not
+random permutation fixtures or a quotient that identifies owners.
+
+For s != t among those pairs, let C=B_s B_t*. A nonidentity regular
+translation has trace zero. Distinct relative routes imply that the
+identity coefficient of C C* receives contributions only from equal
+port indices. Its eight-by-eight matrix is
+
+    sum_p (r_p r_p^T)^2
+      = sum_p ||r_p||^2 r_p r_p^T
+      = 4 R_U R_U^T
+      = 192 I_8.
+
+Therefore the full regular trace is 1536N, not 192N. Define orthogonal
+projections on the ordinary-coordinate space by
+
+    Q_s = B_s* B_s / 48.
+
+Their traces satisfy
+
+    tr Q_s = 8N,
+    tr(Q_s Q_t) = 1536N / 48^2 = (2/3)N.
+
+For either triple I={0,1,2} or I={2,3,4}, set S_I=sum_(s in I) Q_s.
+It is positive semidefinite, and
+
+    tr S_I = 24N,
+    tr(S_I^2) = 3(8N) + 6(2N/3) = 28N.
+
+Cauchy--Schwarz on its nonzero eigenvalues gives
+
+    rank S_I >= (tr S_I)^2 / tr(S_I^2) = 144N/7.       (1)
+
+Its kernel is the common kernel of the three B_s, so (1) is also a
+lower bound on the rank of the stacked three-reader map. Rational and
+real matrix ranks agree. The trace argument is a proof for the actual
+finite regular action; the checker certifies its finite route and
+coefficient premises without enumerating G.
+
+## 3. A nonzero compatible pair of ordinary inputs exists
+
+Let the two subspaces of the 8N-dimensional query space be
+
+    V = B1(ker B0 intersect ker B2),
+    W = B2(ker B3 intersect ker B4).
+
+For linear maps A and B on the same domain,
+
+    dim B(ker A) = rank [A;B] - rank A.
+
+Applying (1), and bounding each two-reader rank above by 16N, gives
+
+    dim V >= ceil(32N/7),
+    dim W >= ceil(32N/7).                            (2)
+
+The transport M=-ell_(2,A) P_(1,A) is invertible between the two query
+spaces: both factors are selected-coordinate isomorphisms. Hence
+
+    dim(M V intersect W)
+      >= dim V + dim W - 8N
+      >= 2 ceil(32N/7) - 8N
+      >= ceil(8N/7).                                (3)
+
+For every nonzero w in this intersection choose ordinary vectors z0,z1
+with
+
+    B0 z0 = B2 z0 = 0,
+    B3 z1 = B4 z1 = 0,
+    B2 z1 = -ell_(2,A) P_(1,A) B1 z0 = w != 0.       (4)
+
+Set
+
+    d = -P_(1,A) B1 z0,
+
+so `ell_(2,A)d=w`, and d is nonzero. Everything is rational, so finite
+rational linear sections can choose z0,z1 linearly in w. This proves
+existence of an actual regular-cover family; it does not emit a compact
+individual owner vector or assert an efficient native way to prepare it.
+No support-size, coefficient-bitlength or precision bound is supplied.
+
+## 4. The exact two-bank trajectory
+
+Initialize pair 0 by
+
+    X_(0,U)=z0,             Y_(0,U)=0,
+    X_(0,A)=d,              Y_(0,A)=-d,
+    X_(0,T)=-P_(0,T) ell_(0,A)d,
+    Y_(0,T)=-X_(0,T).
+
+Initialize pair 1 with `X_(1,U)=z1` and every other field zero. All
+fields of pairs 2 and 3 are zero. The sixteen selected pair sums are
+zero in every pair. All eight initial query records vanish by (4).
+
+At F0, `f0=B0 z0+ell_(0,A)d-ell_(0,A)d=0`. The selected output is
+`X_A=-d,Y_A=d,X_T=Y_T=0`; the ordinary pair is `(z0,z0)`.
+
+At B_(1,2), the future A logger leaves d unchanged because `B2 z0=0`.
+The old T driver is zero. Its current query is
+
+    b1 = B1 z0 + ell_(1,A)d = 0.
+
+The state immediately after B1 is therefore
+
+    pair 0 ordinary: (X_U,Y_U)=(0,z0),
+    pair 0 selected: (X_A,Y_A)=(-d,d), X_T=Y_T=0,
+    pair 1 ordinary: (z1,0), all its selected fields zero.       (5)
+
+The first positive bridge adds z1 to X_(0,U) and subtracts z0 from
+Y_(1,U). At F2 the full query is
+
+    f2 = B2 z1 - ell_(2,A)d = 0.
+
+The forward rule moves the zero selected T value into A and writes the
+zero query into T; conserved sums make every selected Y value zero too.
+After the following negative bridge, all selected fields are zero and
+
+    pair 0 ordinary: (0,z0+z1),
+    pair 1 ordinary: (z1,z1).                         (6)
+
+At B_(3,4), the future query is `B4 z1=0`, the old T driver is zero,
+and the current query is `B3 z1=0`. It follows that pair 1 becomes
+`(0,z1)` with all selected fields zero.
+
+The remaining positive and negative bridge around F4 cancels its
+transient Y_(2,U) value. Every later active local source is zero, so
+f4,f6,f8 are zero and every selected field stays zero. The final three
+ordinary bridges leave, immediately before the terminal,
+
+    all X_(k,U)=0,
+    Y_(0,U)=z0, Y_(1,U)=z1, Y_(2,U)=Y_(3,U)=0.
+
+After `(X,Y)->(Y,-X)`, the only possibly nonzero fields are the two
+former-Y ordinary endpoints, final X_(0,U)=z0 and final X_(1,U)=z1.
+Every allowed observation is zero, but the required original X_(0,A)=d
+is nonzero.
+
+The verifier implements the local maps literally in a free ordered
+linear-expression algebra, using only the selected inverse identities
+and relations (4). It binds every event to the archive recipe and
+records all nineteen intermediate ordinary states. Its universal force
+comes from the written derivation and actual-source existence argument,
+not a finite random model.
+
+For a linear section of the intersection in (3), the original A input
+d is an injective function of w. Thus J maps this zero-observation
+family to a space of dimension at least ceil(8N/7), proving the theorem.
+
+## 5. Scope, controls, and remaining obligations
+
+After complete F2 every selected field is zero. Accordingly the same
+family remains invisible if arbitrary homogeneous selected-only linear
+operations are inserted after that cut, without extra ordinary reads or
+changes to the retained ordinary schedule. This extension does not allow
+an earlier selected swap, affine input-dependent auxiliary state, or a
+record of d at B1. A B1-cut record and the initial B1 X_(0,U) query both
+detect the family, as the rejecting controls demonstrate.
+
+The primary checker rejects changed centre coefficients, collided owner
+routes, wrong initial T sign, wrong selected-transport sign, wrong
+cross-bank link sign, omitted B3/B4 kernel premises, a missing negative
+bridge, wrong local pair/orientation or bridge sign, a native route placed
+at the wrong port with its registry unchanged, and observations that incorrectly treat former-Y endpoints or
+the B1 cut as available. The independent checker and package wrapper
+provide separate evidence described in README.md.
+
+No full native word, supplier invoice, hosting or precision theorem is
+claimed or replayed here. The original local native admission is
+inherited. The result does not establish the selected multiplication
+frontier, optimize the transcript dimension, or rule out other choices
+of 64 initial query coordinates. Its useful consequence is narrower:
+the specified bank-local records do not convert the recorded-cut
+recovery into a final-only recovery on this actual routed schedule.
+
+## Credit
+
+Ali Romman (PR #1969) supplies the QQ16 local maps and retained packet.
+The original owner embedding, centre geometry and selected recipe retain
+their supplier and compiler lineage. Ross Richardson (PR #2104) supplies
+the trace/rank method; @commdata2338 (PR #2174) supplies the explicitly
+recorded-cut two-query recovery contract. Philippe Van Velzen's
+`680d0997621a2cecfef2186ab143377658687825` four-bank experiment supplies
+the earlier final-only finite-model failure and the all-bank record
+choice. Adam Gall (PR #2093) supplies adjacent actual-route hidden-query
+analysis. This contribution proves a new actual-cover two-bank
+intersection obstruction for that final-only contract; it does not claim
+those predecessor observations or methods as new.
+
+Prepared with substantial OpenAI Codex assistance. Apache-2.0.